What IS a complex factor?

I know how to factorize a polynomial over C, but what do complex factors show? Real factors show where the graph cuts the x-axis. I know how to do the calculations and pass the tests, but they never actually explain these type of things in text books. For example:
z3-8z2+25z-26 = (z-2)(z-3-2i)(z-3+2i)
The graph cuts the x-axis at z (or x) = 2, as expected, but what do the complex factors show?

For a normal single-variable function ##f(x)## you have one independent variable, x, and one dependent variable, ##f##, which would be plotted using the y-axis. You give the function a single value of x, and it returns a single value of ##f(x)##.

When you are dealing with a complex function, i.e. ##f(z)##, you are really looking at a function of TWO independent variables, a and b. So, ##f(z)=f(a+ib)=f(z(a, b))##. The function ##f(z)## can't return any information unless you specify both a AND b. In other words, you have to give a fully defined complex number, which requires defining the real part AND the imaginary part.

When you graph a single variable function, you have one dimension which represents the domain (i.e. the x axis). Right to left represents the domain, while up and down represents the outputs of your function, or the range. This type of graph cannot work for a complex function, because a complex function has a two dimensional domain.

Do you know how to plot a complex number on the complex plane? Imagine slanting the complex plane so that it's horizontal, this is what the domain of a complex function ##f(z)## looks like. You can pick any number ##a+ib## and you can put it into ##f(z)##, simplify it, and get another complex number out. So do you see what I mean by the domain of a complex function being two dimensional? When you see a function ##f(x)## we are automatically assuming that this function doesn't take on complex numbers. The fact that it's listed as a single variable function implies that we only accept real values in the function. With this restriction, we can consider the domain being one dimensional and we can plot the function using a normal cartesian coordinate system. Unfortunately, if we place this restriction on the domain, we don't see the whole story. In particular, we can't see all the zeroes, we can only see the real zeroes.

If you were able to plot a complex function entirely so that its whole domain is visible, you would see that it has zeroes at all those points on the complex plane. For example, if ##f(z)## has a zero at ##z=\alpha+i\beta## then that means that ##f(\alpha+i\beta)=0##. Unfortunately, you can't really fully graph a complex function though because the domain is two dimensional and the range is also two dimensional, this means that the graph of a complex function exists in 4D space. There are ways to fully graph a complex function, but they can be quite confusing to interpret.

Whether or not the complex roots say anything about the REAL graph is another question though.